Representation Theory and Noncommutative Geometry, Paris

Reciprocal hyperbolic elements in PSL2(Z)

5 févr. 2025, 11:15

55m

Amphitheater Darboux (IHP)

Orateur

Alain Valette (Université de Neuchâtel)

Description

An element $A$ in ${\mathrm{PSL}}_2(\mathbb{Z})$ is hyperbolic if $|\mathrm{Tr}(A)|>2$. The maximal virtually abelian subgroup of ${\mathrm{PSL}}_2(\mathbb{Z})$ containing $A$ is either infinite cyclic or infinite dihedral; say that $A$ is reciprocal if the second case happens ($A$ is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in ${\mathrm{PSL}}_2(\mathbb{Z})$ in terms of the continued fractions of their fixed points in ${\mathrm{P}}^1(\mathbb{R})$ (those are quadratic surds). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke \& Klein.

https://sites.google.com/view/julgfest